I'm a little lost on the dog subplot. The chief witnesses someone shooting dogs. Everyone thinks the chief is shooting the dogs. The chief reports a guy in a black pickup truck is doing it. The chief's father hallucinates.
The dog killer shows up and speaks to the chief and his daughter, handing her a pack of beer.…

The hand print reminded me of the mini dome thing where the four of them had to touch it.
Did the time traveling girl just replace the other girl through the hand print?
I'm probably over thinking this.

Brienne met Arya, so I am not sure she would just leave the area without spending time looking for her.
Brienne never really wanted to fight the Hound, he forced it. She should at least try to get more info out of him. She knows exactly where he fell down the hill.
That being said, Arya and the Hound left the Eyrie…

Wouldn't the bells toll for a prisoner escape? especially one scheduled for execution.

IMHO, he knows the truth. However, to quote Littlefinger "things only become real when you say it out loud"
If you talk about Cersei and Jaime then Tommen isn't the king. So don't talk about it.
Speak aloud what is in your best interest and push people out the moon door if they say things that are not good for you.

Didn't her adopted father state that Red was her biological father. Red hurried his death so that he couldn't tell her.

After she lets Ward die, she is still trapped. Which leads to more torture.